Unfolding Convex Polyhedra via Quasigeodesics
description
Transcript of Unfolding Convex Polyhedra via Quasigeodesics
![Page 1: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/1.jpg)
Unfolding Convex Unfolding Convex PolyhedraPolyhedra
via Quasigeodesicsvia Quasigeodesics
Jin-ichi Ito (Kumamoto Univ.)Jin-ichi Ito (Kumamoto Univ.)
Joseph O’Rourke (Smith Joseph O’Rourke (Smith College)College)
Costin VCostin Vîîlcu (S.-S. Romanian lcu (S.-S. Romanian Acad.)Acad.)
![Page 2: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/2.jpg)
TheoremTheorem: Every : Every convexconvex polyhedron polyhedron has a has a generalgeneral nonoverlapping nonoverlapping unfolding (a net).unfolding (a net).
General Unfoldings of General Unfoldings of Convex PolyhedraConvex Polyhedra
Source unfolding [Sharir & Schorr ’86, Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]Mitchell, Mount, Papadimitrou ’87]
Star unfolding [Aronov & JOR ’92]Star unfolding [Aronov & JOR ’92]
[Poincare 1905?]
![Page 3: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/3.jpg)
Shortest paths from Shortest paths from xx to all to all verticesvertices
[Xu, Kineva, O’Rourke 1996, 2000]
![Page 4: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/4.jpg)
![Page 5: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/5.jpg)
Source UnfoldingSource Unfolding
![Page 6: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/6.jpg)
Star UnfoldingStar Unfolding
![Page 7: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/7.jpg)
Star-unfolding of 30-vertex Star-unfolding of 30-vertex convex polyhedronconvex polyhedron
![Page 8: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/8.jpg)
TheoremTheorem: Every : Every convexconvex polyhedron polyhedron has a has a generalgeneral nonoverlapping nonoverlapping unfolding (a net).unfolding (a net).
General Unfoldings of General Unfoldings of Convex PolyhedraConvex Polyhedra
Source unfolding Source unfolding Star unfolding Star unfolding Quasigeodesic unfoldingQuasigeodesic unfolding
![Page 9: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/9.jpg)
Geodesics & Closed Geodesics & Closed GeodesicsGeodesics
GeodesicGeodesic: locally shortest path; : locally shortest path; straightest lines on surfacestraightest lines on surface
Simple geodesicSimple geodesic: non-self-: non-self-intersectingintersecting
Simple, Simple, closed geodesicclosed geodesic:: Closed geodesic: returns to start w/o Closed geodesic: returns to start w/o
cornercorner (Geodesic loop: returns to start at (Geodesic loop: returns to start at
corner)corner)
![Page 10: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/10.jpg)
Lyusternick-Schnirelmann Lyusternick-Schnirelmann TheoremTheorem
TheoremTheorem: Every closed surface : Every closed surface homeomorphic to a sphere has at homeomorphic to a sphere has at least three, distinct closed least three, distinct closed geodesics.geodesics.
![Page 11: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/11.jpg)
QuasigeodesicQuasigeodesic
Aleksandrov 1948Aleksandrov 1948 left(p) = total incident face angle left(p) = total incident face angle
from leftfrom left quasigeodesic: curve s.t. quasigeodesic: curve s.t.
left(p) ≤ left(p) ≤ right(p) ≤ right(p) ≤
at each point p of curve.at each point p of curve.
![Page 12: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/12.jpg)
Closed QuasigeodesicClosed Quasigeodesic
[Lysyanskaya, O’Rourke 1996]
![Page 13: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/13.jpg)
![Page 14: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/14.jpg)
Shortest paths to Shortest paths to quasigeodesic do not touch quasigeodesic do not touch
or crossor cross
![Page 15: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/15.jpg)
![Page 16: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/16.jpg)
Insertion of isosceles Insertion of isosceles trianglestriangles
![Page 17: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/17.jpg)
Unfolding of CubeUnfolding of Cube
![Page 18: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/18.jpg)
![Page 19: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/19.jpg)
![Page 20: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/20.jpg)
ConjectureConjecture
BaseBase Source Source UnfoldingUnfolding
Star Star UnfoldingUnfolding
pointpoint theoremtheorem theoremtheorem
quasigeodesicquasigeodesic ?? theoremtheorem
![Page 21: Unfolding Convex Polyhedra via Quasigeodesics](https://reader035.fdocuments.in/reader035/viewer/2022081504/56813dcc550346895da794a4/html5/thumbnails/21.jpg)
Open: Find a Closed Open: Find a Closed QuasigeodesicQuasigeodesic
Is there an algorithmIs there an algorithmpolynomial timepolynomial time
or efficient numerical algorithmor efficient numerical algorithm
for finding a closed quasigeodesic on a for finding a closed quasigeodesic on a (convex) polyhedron?(convex) polyhedron?